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Consider the 3D Gaussian distribution $(x,\tilde{\pi}(x), \pi(\tilde{\pi}(x)))\sim\mathcal{N}(\mu, \Sigma)$ where $\tilde{\pi}(x)=\psi x+v$ and $\pi(\tilde{\pi}(x))=u\sin(\tilde{\pi}(x))$. $\psi$, v and u are constant values.

I want to marginalize out $\tilde{\pi}(x)$ in order to obtain marginal distribution $f_{x,\pi}$. I know in a multivariate distribution I should integrate out the unwanted variables to obtain the marginal distribution.

But since $\tilde{\pi}(x)$ is a function of x and $\pi(\tilde{\pi}(x))$ is a function of $\tilde{\pi}$, I'm not sure how I'm supposed to find the marginal distribution$f_{x,\pi}$?

Thanks for your help.

Regards.

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    Is $\Sigma$ known? – 2017-01-19
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    Yes the normal distribution mean and covariance matrix are known. – 2017-01-19
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    then the dependencies between each entry are not needed. The information of the joint distribution is enough. – 2017-01-19
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    Thanks for your reply. So you mean I just need to compute the marginal distribution regardless of the dependencies? Just integrate out the unwanted variable without substitution of the relations mentioned above, Right? Please correct me if I'm wrong. Thanks again. – 2017-01-19
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    you are welcome. Yes, there is an explicit formula for this actually. – 2017-01-19

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So according to Vim comments, I think here's how I should find the marginal distribution:

\begin{equation} f(x, \pi)=\int_{\psi x_{min} + v}^{\psi x_{max}+v} f(x, \tilde{\pi}, \pi) d \tilde{\pi} \end{equation}

I set the lower and upper bounds on the integral according to the relation between $\tilde{\pi}(x)$ and x. Again any more comments are welcome in case I'm wrong. Thanks again.